Problem: Determine how many solutions exist for the system of equations. ${-2x+y = -7}$ ${4x-2y = 14}$
Answer: Convert both equations to slope-intercept form: ${-2x+y = -7}$ $-2x{+2x} + y = -7{+2x}$ $y = -7+2x$ ${y = 2x-7}$ ${4x-2y = 14}$ $4x{-4x} - 2y = 14{-4x}$ $-2y = 14-4x$ $y = -7+2x$ ${y = 2x-7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x-7}$ ${y = 2x-7}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-2x+y = -7}$ is also a solution of ${4x-2y = 14}$, there are infinitely many solutions.